is a para-symmetric strongly minimal linearization of \(P(\lambda )\) . It is obvious that \(P(\lambda )\)  and \(\widehat{L}_s (\lambda )\)  have the same finite eigenvalues with the same partial multiplicities because both have \(\lambda = 0\)  as unique finite eigenvalue with partial multiplicity 2. Observe that \(P(\lambda )\)  has also an eigenvalue at infinity with partial multiplicity 2. This infinite structure can be recovered from \(\widehat{L}_s (\lambda )\)  by using Theorem REF  or, better, the simpler Corollary REF  specific for quadratic polynomials. For this purpose, note that, with the notation of Corollary REF , \(r_P = 2\)  and \(r_2 = 1\) . Moreover, \(\mathrm {rev}_1 \widehat{L}_s (\lambda )\)  has only one partial multiplicity at 0 equal to 1. So, \(u = 1\)  in Corollary REF , which yields that \(\mathrm {rev}_2 P(\lambda )\)  has 2 as its unique partial multiplicity at 0. Note that \(P(\lambda )\)  in (REF ) is also symmetric. It is easy to construct infinitely many symmetric GLR-strong linearizations of \(P(\lambda )\)  [1]}. All of them have necessarily size \(4 \times 4\) . In contrast, the symmetric strongly minimal linearization constructed by using Theorem  has size \(3 \times 3\) .

